Respuesta :
Answer:
read below.
Step-by-step explanation:
To find the value of \(9x^2 + 16y^2\), we can first manipulate the given equations to isolate either \(x\) or \(y\) and then substitute the result into the expression for \(9x^2 + 16y^2\).
Given:
1. \(3x + 4y = 12\) ...(1)
2. \(xy = 4\) ...(2)
From equation (2), we can express \(x\) in terms of \(y\) by dividing both sides by \(y\):
\[ x = \frac{4}{y} \] ...(3)
Now, substitute the expression for \(x\) from equation (3) into equation (1):
\[ 3 \left( \frac{4}{y} \right) + 4y = 12 \]
Simplify:
\[ \frac{12}{y} + 4y = 12 \]
Multiply both sides by \(y\) to clear the fraction:
\[ 12 + 4y^2 = 12y \]
Rearrange the terms:
\[ 4y^2 - 12y + 12 = 0 \]
Divide both sides by 4 to simplify:
\[ y^2 - 3y + 3 = 0 \]
Now, we can use the quadratic formula to solve for \(y\):
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = -3\), and \(c = 3\).
\[ y = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(3)}}{2(1)} \]
\[ y = \frac{3 \pm \sqrt{9 - 12}}{2} \]
\[ y = \frac{3 \pm \sqrt{-3}}{2} \]
Since the discriminant is negative, the solutions will be complex. Let's denote the solutions as \(y_1\) and \(y_2\):
\[ y_1 = \frac{3 + i\sqrt{3}}{2} \]
\[ y_2 = \frac{3 - i\sqrt{3}}{2} \]
Now, substitute each value of \(y\) into equation (3) to find the corresponding values of \(x\):
\[ x_1 = \frac{4}{y_1} = \frac{4}{\frac{3 + i\sqrt{3}}{2}} = \frac{8}{3 + i\sqrt{3}} \]
\[ x_2 = \frac{4}{y_2} = \frac{4}{\frac{3 - i\sqrt{3}}{2}} = \frac{8}{3 - i\sqrt{3}} \]
Now, we can calculate \(9x^2 + 16y^2\) using both sets of solutions for \(x\) and \(y\), and then summing the results:
For \(x_1\) and \(y_1\):
\[ 9x_1^2 + 16y_1^2 = 9\left( \frac{8}{3 + i\sqrt{3}} \right)^2 + 16\left( \frac{3 + i\sqrt{3}}{2} \right)^2 \]
For \(x_2\) and \(y_2\):
\[ 9x_2^2 + 16y_2^2 = 9\left( \frac{8}{3 - i\sqrt{3}} \right)^2 + 16\left( \frac{3 - i\sqrt{3}}{2} \right)^2 \]
Calculating these expressions will give us the value of \(9x^2 + 16y^2\) using both sets of solutions for \(x\) and \(y\).
Answer:
9x² + 16y² = 48
Step-by-step explanation:
given 3x + 4y = 12 and xy = 4
(3x + 4y)² ← expand using FOIL
(3x + 4y)² = 9x² + 24xy + 16y² ← substitute given values
12² = 9x² + 24(4) + 16y²
144 = 9x² + 96 + 16y² ( subtract 96 from both sides )
48 = 9x² + 16y²
